Question

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x+y+z$ $=9$ at point Q. If the length of the line segment PQ is $l$, then value of $l^2$ is

Answer 1

The line makes equal angles with axes

$⟹l=m=n=\frac{1}{\sqrt{3}}$
Given plane equation $2x+y+z$ $=$ $9$
$\therefore$ equations of line are $\frac{x-2}{1/\sqrt{3}}=\frac{y+1}{1/\sqrt{3}}=\frac{z-2}{1/\sqrt{3}}$
$x-2=y+1=z-2=r$
$Q\equiv (r+2,r-1,r+2)$
$\because$ Q Lies on the plane $2x+y+z$ $=$ $9$
$2(r+2)+(r-1)+(r+2)$ $=$ $9$
$\Rightarrow 4r+5=9\Rightarrow$ r $=$ 1
$\therefore$ $Q(3,0,3)$
$\therefore PQ=\sqrt{1+1+1}=\sqrt{3}$